## Exploding Dots

### 1.3 A Peek at a Weird Machine

Lesson materials located below the video overview.

Let’s get quirky! Here are two first attempts at weirdness:

**What do you think of a \(1 \leftarrow 1\)** **machine? **

What happens if you put in a single dot? Is a \(1 \leftarrow 1\) machine interesting? Helpful?

**What do you think of a \(2 \leftarrow 1\)** **machine? **

What happens if you put in a single dot?

What do you think of the utility of a \(2 \leftarrow 1\) machine?

** **

Okay … How about this then?

**Consider the \(2 \leftarrow 3\) machine. **

*This machine replaces three dots in one box with two dots one place to their left. *

This machine seems to do interesting things. For example, placing ten dots into the machine produces the following result:

### \(10 \rightarrow 2101\)

(Check this!)

Here are the \(2 \leftarrow 3\) codes for the first fifteen numbers:

Hmm. Do these mean anything?

Some questions:

*Does it make sense that only the digits \(0\), \(1\), and \(2\) appear in these codes?*

*Do all codes after then number one begin with \(2\)? Begin with \(21\) (from six onwards)?*

*Does it make sense that the final digits of these codes cycle \(1, 2, 0, 1, 2, 0, 1, 2, 0, ….\)?*

*Can one do arithmetic in this weird system? For example, here is what \(6+5\) looks like. Is the answer indeed eleven?*

(Actually, maybe the real answer is \(402\) or better \(2102\). Do you see what I am thinking? Is this the proper code for eleven?)

But the real question is:

**What are these codes? What are we doing representing numbers this way?**

**Are these codes for numbers written in base three? In base two? In some other base? **

There are a lot of shockers to be had about this \(2 \leftarrow 3\) machine. We’ll explore this machine in detail in section 3.1 of this course, and present a number of open research questions about it.

For now, just play with the \(2 \leftarrow 3\) machine and get a feel for it. Perhaps write out the codes for all the numbers up to fifty and look for structure and patterns. And think about the question: *Are we writing numbers in some base? If so, what base?* More on this, for sure, later!

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